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\section*{duality between M/F-theory and heterotic string theory}

\begin{quote}%
This entry is about [[M-theory]]/[[F-theory]] [[KK-compactification|compactified]] on [[K3-surfaces]]. For [[M-theory]] on [[MO9]]-planes see instead at \emph{[[Hořava-Witten theory]]}.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{duality_in_string_theory}{}\paragraph*{{Duality in string theory}}\label{duality_in_string_theory}

[[!include duality in string theory -- contents]]

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{aspects}{Aspects}\dotfill \pageref*{aspects} \linebreak
\noindent\hyperlink{from_mbranes_to_fbranes_to_heterotic_strings_and_ns5branes}{From M-branes to F-branes to heterotic strings and NS5-branes}\dotfill \pageref*{from_mbranes_to_fbranes_to_heterotic_strings_and_ns5branes} \linebreak
\noindent\hyperlink{NonReducibleHeteroticGaugeBackgrounds}{Non-reducible heterotic $E_8$-gauge backgrounds}\dotfill \pageref*{NonReducibleHeteroticGaugeBackgrounds} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{for_type_iia_and_mtheory}{For type IIA and M-theory}\dotfill \pageref*{for_type_iia_and_mtheory} \linebreak
\noindent\hyperlink{for_ftheory}{For F-theory}\dotfill \pageref*{for_ftheory} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A [[duality in string theory]]. The [[non-perturbative effect|non-perturbative]] enhancement of [[duality between heterotic and type II string theory]]:

[[F-theory]]$\,$ ``[[KK-compactification|KK-compactified]]'' on an [[elliptic fibration|elliptically fibered]] [[K3]] with a [[section]] is supposed to be equivalent to [[heterotic string theory]] [[KK-compactification|KK-compactified]] on a [[2-torus]].

More generally, [[F-theory]] on a [[complex manifold|complex]] $n$-dimensional $X$ fibered $X\to B$ with elliptic [[K3]]-[[fibers]] is supposed to be equivalent to [[heterotic string theory]] on an [[elliptic fibration|elliptically fibered]] [[Calabi-Yau manifold]] $Z \to B$ of [[complex manifold|complex]] [[dimension]] $(n-1)$.

A detailed discussion of the [[equivalence]] of the respective [[moduli spaces]] is originally due to (\hyperlink{FriedmanMorganWitten97}{Friedman-Morgan-Witten 97}). A review of this is in (\hyperlink{Donagi98}{Donagi 98}).

From the abstract of (\hyperlink{Donagi98}{Donagi 98}).

\begin{quote}%
The [[heterotic string theory|heterotic string]] [[KK-compactification|compactified]] on an $(n-1)$-[[dimension|dimensional]] [[elliptic fibration|elliptically fibered]] [[Calabi-Yau variety|Calabi-Yau]] $Z \to B$ is conjectured to be [[duality in string theory|dual]] to [[F-theory]] [[KK-compactification|compactified]] on an $n$-dimensional [[Calabi-Yau variety|Calabi-Yau]] $X \to B$, fibered over the same base with [[elliptic fibration|elliptic]] [[K3]] fibers. In particular, the [[moduli]] of the two theories should be [[isomorphism|isomorphic]]. The cases most relevant to the physics are $n=2$, $3$, $4$, i.e. the [[KK-compactification|compactification]] is to [[dimensions]] $d=8$, $6$ or $4$ respectively. Mathematically, the richest picture seems to emerge for $n=3$, where the [[moduli space]] involves an analytically [[integrable system]] whose fibers admit rather different descriptions in the two theories.


\end{quote}
\hypertarget{aspects}{}\subsection*{{Aspects}}\label{aspects}

\hypertarget{from_mbranes_to_fbranes_to_heterotic_strings_and_ns5branes}{}\subsubsection*{{From M-branes to F-branes to heterotic strings and NS5-branes}}\label{from_mbranes_to_fbranes_to_heterotic_strings_and_ns5branes}

[[!include F-branes -- table]]

\hypertarget{NonReducibleHeteroticGaugeBackgrounds}{}\subsubsection*{{Non-reducible heterotic $E_8$-gauge backgrounds}}\label{NonReducibleHeteroticGaugeBackgrounds}

There are some [[F-theory]] backgrounds whose supposed dual in [[heterotic string theory]] involves an [[E8]]-[[principal connection]] which is not [[reduction of the structure group|reducible]] to [[SemiSpin(16)]] (\hyperlink{DistlerSharpe10}{Distler-Sharpe 10, section 5}), while in fact the traditional construction of the heterotic [[worldsheet]] theory only covers this case. In (\hyperlink{DistlerSharpe10}{Distler-Sharpe 10, section 7-8}) it is argued that therefore a more general formulation of [[heterotic string theory]] needs to involve [[parameterized WZW models]]. See also at \emph{\href{http://ncatlab.org/nlab/show/heterotic+string+theory#GeneralGaugeBackgroundsAndParameterizedWZWModels}{heterotic string -- Properties -- General gauge backgrounds and parameterized WZW models}}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include F-theory compactifications -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{for_type_iia_and_mtheory}{}\subsubsection*{{For type IIA and M-theory}}\label{for_type_iia_and_mtheory}

The conjectured duality between [[type IIA string theory]] [[KK-compactification|KK-compactified]] on [[K3]] times an [[n-torus]] and [[heterotic string theory]] on the $(n+2)$-torus is originally due to

\begin{itemize}%
\item [[Chris Hull]], [[Paul Townsend]], section 6 of \emph{Unity of Superstring Dualities}, Nucl.Phys.B438:109-137,1995 (\href{http://arxiv.org/abs/hep-th/9410167}{arXiv:hep-th/9410167})


\item [[Edward Witten]], section 4 of \emph{[[String Theory Dynamics In Various Dimensions]]}, Nucl.Phys.B443:85-126,1995 (\href{http://arxiv.org/abs/hep-th/9503124}{arXiv:hep-th/9503124})



\end{itemize}
Review includes

\begin{itemize}%
\item [[Paul Aspinwall]], [[David Morrison]], \emph{String Theory on K3 Surfaces}, in [[Brian Greene]], [[Shing-Tung Yau]] (eds.), \emph{Mirror Symmetry II}, International Press, Cambridge, 1997, pp. 703-716 (\href{https://arxiv.org/abs/hep-th/9404151}{arXiv:hep-th/9404151})


\item [[Paul Aspinwall]], \emph{K3 Surfaces and String Duality}, in [[Shing-Tung Yau]] (ed.): \emph{Differential geometry inspired by string theory} 1-95 (\href{https://arxiv.org/abs/hep-th/9611137}{arXiv:9611137} \href{http://inspirehep.net/record/426102}{spire:426102})



\end{itemize}
Further discussion includes

\begin{itemize}%
\item [[Paul Aspinwall]], \emph{Enhanced Gauge Symmetries and K3 Surfaces}, Phys.Lett. B357 (1995) 329-334 (\href{http://arxiv.org/abs/hep-th/9507012}{arXiv:hep-th/9507012})


\item [[Eric Bergshoeff]], C. Condeescu, G. Pradisi, F. Riccioni, \emph{Heterotic-Type II duality and wrapping rules}, JHEP12(2013)057 (\href{https://arxiv.org/abs/1311.3578}{arXiv:1311.3578})



\end{itemize}
Specifically in relation to the putative [[K-theory]]-classification of [[D-brane charge]]:

\begin{itemize}%
\item Inaki Garcia-Etxebarria, [[Angel Uranga]], \emph{From F/M-theory to K-theory and back}, JHEP 0602:008,2006 (\href{https://arxiv.org/abs/hep-th/0510073}{arXiv:hep-th/0510073})

\end{itemize}
Specifically in [[M-theory on G2-manifolds]]:

\begin{itemize}%
\item [[Michael Atiyah]], [[Edward Witten]] section 6.4 of \emph{$M$-Theory dynamics on a manifold of $G_2$-holonomy}, Adv. Theor. Math. Phys. 6 (2001) (\href{http://arxiv.org/abs/hep-th/0107177}{arXiv:hep-th/0107177})

\end{itemize}
Specifically in relation to [[Moonshine]]:

\begin{itemize}%
\item [[Miranda Cheng]], Sarah M. Harrison, Roberto Volpato, Max Zimet, \emph{K3 String Theory, Lattices and Moonshine} (\href{https://arxiv.org/abs/1612.04404}{arXiv:1612.04404})

\end{itemize}
\hypertarget{for_ftheory}{}\subsubsection*{{For F-theory}}\label{for_ftheory}

Discussion for [[F-theory]] includes

\begin{itemize}%
\item [[David Morrison]], [[Cumrun Vafa]], \emph{Compactifications of F-Theory on Calabi--Yau Threefolds -- I}, Nucl.Phys. B473 (1996) 74-92 (\href{http://arxiv.org/abs/hep-th/9602114}{arXiv:hep-th/9602114})


\item [[David Morrison]], [[Cumrun Vafa]], \emph{Compactifications of F-Theory on Calabi--Yau Threefolds -- II}, Nucl.Phys.B476:437-469,1996 (\href{http://arxiv.org/abs/hep-th/9603161}{arXiv:hep-th/9603161})


\item [[Ashoke Sen]], \emph{F-theory and Orientifolds} (\href{http://arxiv.org/abs/hep-th/9605150}{arXiv:hep-th/9605150})


\item Robert Friedman, [[John Morgan]], [[Edward Witten]], \emph{Vector Bundles And F Theory}, Commun.Math.Phys.187:679-743, 1997 (\href{http://arxiv.org/abs/hep-th/9701162}{arXiv:hep-th/9701162})


\item [[Paul Aspinwall]], \emph{M-Theory Versus F-Theory Pictures of the Heterotic String}, Adv.Theor.Math.Phys.1:127-147, 1998 (\href{http://arxiv.org/abs/hep-th/9707014}{arXiv:hep-th/9707014})



\end{itemize}
Review of (\hyperlink{FriedmanMorganWitten97}{Friedman-Morgan-Witten 97}) is in

\begin{itemize}%
\item [[Ron Donagi]], \emph{ICMP lecture on heterotic/F-theory duality} (\href{http://arxiv.org/abs/hep-th/9802093}{arXiv:hep-th/9802093})


\item Bj\"o{}rn Andreas, \emph{$N=1$ Heterotic/F-theory duality}, PhD thesis (\href{http://edoc.hu-berlin.de/dissertationen/physik/andreas-bjoern/PDF/Andreas.pdf}{pdf})



\end{itemize}
with more details in

\begin{itemize}%
\item [[Ron Donagi]], [[Eyal Markman]], \emph{Spectral curves, algebraically completely integrable Hamiltonian systems, and moduli of bundles} (\href{http://arxiv.org/abs/alg-geom/9507017}{arXiv:alg-geom/9507017})

\end{itemize}
The issue with non-reducible $E_8$-gauge connections is highligted in

\begin{itemize}%
\item [[Jacques Distler]], [[Eric Sharpe]], section 5 of \emph{Heterotic compactifications with principal bundles for general groups and general levels}, Adv. Theor. Math. Phys. 14:335-398, 2010 (\href{http://arxiv.org/abs/hep-th/0701244}{arXiv:hep-th/0701244})

\end{itemize}
On a subtlety in the application of the [[Narasimhan-Seshadri theorem]] in the duality:

\begin{itemize}%
\item Herbert Clemens, Stuart Raby, \emph{Heterotic/F-theory Duality and Narasimhan-Seshadri Equivalence} (\href{https://arxiv.org/abs/1906.07238}{arxiv:1906.07238})

\end{itemize}
[[!redirects duality between F-theory and heterotic string theory]]

[[!redirects F/M-theory on elliptically fibered Calabi-Yau 2-folds]]

[[!redirects duality between heterotic string theory and M/F-theory]]



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